Abstract

We study the boundary value problems of second-order singular differential equations. At first, we reduce the BVPs to initial value problems of first-order singular integrodifferential equations and then we employ the quasilinearization method in studying the IVPs and obtain two monotone iterative sequences, which converge uniformly and quadratically to the unique solution of the IVPs. Finally, we get the similar result for the given BVPs.

1. Introduction

It is well known that quasilinearization method is a powerful tool for proving the existence of approximate solutions of nonlinear systems and the converge quadratically to the unique solution of the given problems (see [1]). Recently, Abd-Ellateef Kamar et al. investigated the first-order singular systems of differential equations with initial value problem [2]. In [3], Wang and Liu developed monotone iterative technique combined with the method of upper and lower solutions for studying the second-order singular systems with boundary value problems (BVPs).

In this paper, we extend quasilinearization method to study the second-order singular systems with the boundary conditions:
where is a singular matrix, , , and and are two constant vectors. By using the existence result [4] for linear singular systems and the comparison result [5], we investigate two monotone iterative sequences which converge uniformly and quadratically to the solution of the problem.

2. Preliminaries

Consider the following initial value problem:
where is a singular matrix, , , is an increasing operator, and is a constant vector.

In order to obtain two monotone sequences, we introduce an existence result for the corresponding linear singular systems and a comparison result.

The existence of the solution of the linear initial value problem of the form
is well known and is given by the following lemma.

Lemma 1 (see [4]). Assume that the nonhomogeneous linear system (3) exists and if(i)there exists a such that exists,(ii) is the solution of??, where
and are the Drazin inverses of , respectively, then the solution is given by
Now one gives the following assumptions for convenience.Let and be matrices such that exists and is nonnegative for some . Suppose further that exist and are nonnegative, such that
where , , and is a diagonal square matrix with and .There exist with on , such that
All second-order derivatives of exist and are bounded, is convex in for each , is convex in for each , is nonincreasing in for each , and is nonincreasing in for each .Moreover .To obtain the results, one needs the following comparison theorem.

Lemma 2. Let such that and satisfy assumptions and . Then implies on .The proof is similar to [5] and one omits it.

3. Main Results

Firstly, we develop the following result which is important for the final result.

Theorem 3. Suppose that assumptions hold and, for ,(i), where .
Then there exist two monotone sequences and , which converge uniformly on to the unique solution of problem (2) and the convergence rate is quadratic.

Proof. From we find that , and
for all and . The convexity of in implies that
for , , and .Now consider the following linear problems. For ,
For (10), set
Then, we obtain that . Furthermore, we can get
Hence and are lower and upper solutions of (12), respectively. Thus (12) has a solution on , and we have . Similarly, set
Then we obtain that (14) has a solution on , and we have .Now we claim that
Let . We get that
Noticing that , we get , on .Now, assume that, for , (10) and (11) admit solutions and , respectively, such that
Then setting in (10) and (11), we observe that the assumptions in Theorem 3 are satisfied. Thus, there exist solutions and for (10) and (11), respectively, and we now will show that the following relation
holds. Firstly, we can easily know that and .To prove that , consider . Then
We know that . Hence, from Lemma 2, we deduce that , on . Thus, we have monotone sequences such that
Now, employing Ascoli-Arzela's theorem we conclude that the sequences converge uniformly and monotonically to the unique solution of (2) on .To show that the convergence rate is quadratic, we begin with . Then
where
Set . Then
where , , , and .Next, set . Then
where and .Furthermore, we have that
where .Using Lemma 2, we show that on , where is the solution of
Thus, using Lemma 1, the solution of the previously mentioned equation is given as
After taking suitable estimates, we obtain
where .Set . Then we can get
in which
Set . Then we get that
where and .Similarly, set . Then
where , and .Then we get that
where and . Thus we have that
Furthermore, we obtain after taking suitable estimates
where and .Hence we proved that the convergence rate is quadratic.

Next, we consider singular differential systems BVPs and prove the following main result.

Theorem 4. Let assumptions hold. Suppose further the following.There exist with and on and
For , , where .Then there exist monotone sequences and which converge uniformly on to the unique solution of (1) and the convergence rate is quadratic.

Proof. Using the transformation , we have that
BVPs (1) can be transformed into IVPs of singular system:
Let and . By using assumption , we get that
and .Noticing that , we have that
Then from , and
A similar argument shows that
By Theorem 3, there exist monotone sequences such that
and the convergence rate is quadratic. Again set and . Then
and also the convergence rate is quadratic. Noticing that
we can obtain the similar result that converges quadratically to the solution of (1). The proof is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors completed the paper together. All authors read and approved the final paper.

Acknowledgments

The authors would like to thank the reviewers for their valuable suggestions and comments. This paper is supported by the National Natural Science Foundation of China (11271106) and the Natural Science Foundation of Hebei Province of China (A2013201232).